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The Jonathan Dolhenty Archive |
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INTRODUCTION: Part 6 - Page 1 by Jonathan Dolhenty, Ph.D.
We have learned something about the general and special types of propositions in a previous essay. It's time now to investigate certain properties of these propositions as they are compared to one another. This brings us to the matter of logical opposition. Propositions are said to be logically opposed to each other when they have the same subject and predicate but with a change in quality or quantity or both. The Nature of Logical Opposition We have already learned that all truth is based on the three laws of thought known as the Principle of Identity, the Principle of Contradiction, and the Principle of the Excluded Middle. These three principles are the foundation for all human knowledge. They are self-evident and need no proofs or demonstrations. In fact, they cannot be "proved" in the ordinary sense of the term. If we reject them, however, we are at the end of rational discussion since we need to accept them as true in order to initiate and continue any rational discussion. The Three Laws of Thought There are three important laws of thought that every critical thinker needs to know. Without them, we would find it very difficult to reason correctly. The Principle of Identity The Principle of Identity states that "A is A." Other ways of saying the same thing are "What is, is," "Everything is what it is," and "A thing is identical with itself." Can we seriously challenge such a principle? The Principle of Contradiction The Principle of Contradiction states that "A cannot be A and not A at the same time in the same respect." It can also be stated as "Whatever is, cannot at the same time not be under the same circumstances," or "It is impossible for the same thing both to be and not to be at the same time from the same point of view." From the standpoint of logic, the Principle of Contradiction can be read as "The same attribute cannot at one and the same time be both affirmed and denied of the same thing in the same respect." The Principle of Excluded Middle The Principle of Excluded Middle can be stated in different ways: "A thing either is or is not," "Everything must either be or not be," and "Any attribute must be either affirmed or denied of any given subject." For the purpose of the study of logic, the principle can be stated: "If we make an affirmation, we thereby deny its contradictory; if we make a denial, we thereby affirm its contradictory." Propositional Constructions The logical opposition of propositions is the relation which exists between propositions having the same subject and the same predicate, but differing in quality, or in quantity, or in both. There are four possible way in which a proposition having the same subject and the same predicate may appear: 1. as a universal affirmative (A)
2. as a universal negative (E)
3. as a particular affirmative (I)
4. as a particular negative (O)
These four propositions represent the four types of opposition. They can be diagramed, together with their mutual relations as opposites, in what is called a Square of Opposition. The Traditional Square of Opposition Here are explanations of the four types of opposition and the four relations resulting from the opposition: Subalternation Subalternation: the opposition existing between a universal and particular affirmative (A and I), and between a universal and particular negative (E and O). Both propositions, the universal and the particular, are called subalterns. The universal is the subalternant (A and E). The particular is the subalternate (I and O). Contradiction Contradiction: the opposition existing between a universal affirmative (A) and a particular negative (O), and between a universal negative (E) and a particular affirmative (I). Contrariety Contrariety: the opposition existing between a universal affirmative (A) and a universal negative (E). Subcontrariety Subcontrariety: the opposition existing between a particular affirmative (I) and a particular negative (O). Here is a diagram showing the logical opposition of propositions (with examples provided):
The Laws of Logical Opposition We can now formulate certain laws of truth and falsity regarding propositions which contain these various relations. The Law of Subalternation The Law of Subalternation: A--I and E--O. This law has two phases, depending on whether we begin with the truth or the falsity of one of the subaltern propositions. Beginning with the truth of one of the subaltern propositions (A--I, E--O), the first rule states: The truth of the universal involves the truth of the particular (A to I, E to O); but the truth of the particular does not involve the truth of the universal (I to A, O to E). In other words:
There are, therefore, two sections to this first rule. The first section states: "It is always logical to conclude from the truth of the universal to the truth of the particular." After all, what is true of all individuals of a class must also be true of some of these individuals. What is true of the whole must be true of every part of the whole. Examples: If "All men are mortal," then surely "Some men are mortal." If "No men are dogs," then "Some men are not dogs, either." The second section states: "The truth of the particular does not involve the truth of the universal; the truth of the universal will always be doubtful." What is true of some need not be true of all. What is true of a part of a class need not be true of the whole of the class. Examples: If it is true that "Some men are content," we cannot conclude, on the basis of the proposition alone, that "All men are content." If it is true that "Some men are not content," we cannot conclude that "No men are content." We can see from the examples that the truth of the particular propositions (I and O) does not involve the truth of the universals (A and E). Although the particular propositions I and O are true, their respective universals A and E are false. It could happen, of course, that what is true of some is also true of all and what is true of a part is also true of the whole. In this case, both the particular propositions (I and O) are true, and their respective universals (A and E) are also true. But we are never permitted to conclude from the truth of the particular to the truth of the universal. It may be so, but it need not be so. We cannot validly argue from some to all and from the part to the whole. The second rule of the Law of Subalternation states: The falsity of the particular involves the falsity of the universal; but the falsity of the universal does not involve the falsity of the particular. Here we begin with the falsity of one of the subaltern propositions (I to A, O to E). The rule states:
There are also two sections to this second rule. The first section states: "We can always validly conclude from the falsity of a particular proposition to the falsity of the universal." This makes sense. For something to be true of all, it must be true of every individual that belongs to the all. For something to be true of the whole, it must be true of every part contained in the whole. Example: The particular proposition I, "Some men are dogs," is false. Actually it would be true to say that "Some men are not dogs." In order, however, for the statement to be true that "All men are dogs," it could not be true to say that "Some men are not dogs," because all must include some, and the whole must include every part. Another example: The particular proposition O, "Some men are not mortal," is false. We should say "Some men are are mortal." But if proposition E, "No men are mortal," is true, it would follow that the same some are and are not mortal at the same time. If it is false that "Some men are dogs," it is all the more false to state that "All men are dogs." If it false to say that "Some men are not mortal," it is also false to say that "No men are mortal." From the falsity of the particular proposition (I or O), we must conclude to the falsity of the respective universal proposition (A or E). The second section of this rule states: "If A is false, I need not be false; if E is false, O need not be false." In order that a universal be true, every individual of the class and every part of the whole must be included in the truth of the universal. The universal, therefore, will be false if not every individual of the universal and not every part of the whole is included in the truth of the universal statement. This means that if a universal proposition is false, some of its individuals must also be false, but some of the others may be true. But if some may be true, even if the universal is false, it is obvious we cannot validly conclude from the falsity of the universal to the falsity of the particular. We can now see the truth of the Law of Subalternation. The truth of the universal involves the truth of the particular, but the truth of the particular does not involve the truth of the universal. The falsity of the particular involves the falsity of the universal, but the falsity of the universal does not involve the falsity of the particular. The Law of Contradiction The Law of Contradiction: A--O and E--I. This law has two phases. The first rule states: "Contradictories cannot be true together."
In an affirmative universal (A) proposition, it is asserted that the predicate is affirmed of each and every individual belonging to the subject, as in, for example "All men are mortal." If this is true, then it must be false to deny this statement of some of the individuals. Therefore, the statement that "Some men are not mortal" (O) cannot be true. In a negative universal (E) proposition, it is asserted that the predicate must be denied of each and every individual belonging to the subject, as in, for example "No men are dogs." If this statement is true, then it must be false to say that "Some men are dogs" (I). What is true of all, must be true of every one of the class. To state at the same time that all are and some are not, and that none are and some are, would violate the Principle of Contradiction. The second rule states: "Contradictories cannot be false together."
Example: If it is false that "All men are content," it must be true that "Some men are not content" (A--O). If it is false that "No men are content," it must be true that "Some men are content" (E--I). Another Example: If it is false that "Some men are not mortal," it must be true that "All men are mortal." If it is false that "Some men are dogs," it must true that "No men are dogs." We now can state the following conclusions:
The Law of Contrariety The Law of Contrariety: A--E. There are two rules to be considered. The first rule states that "Contraries cannot be true together." If A is true, E is false and if E is true, A is false. If one of the contraries is true, the other contrary must be false. Example: If "All men are mortal" (a universal affirmative proposition--A) is true, then "No men are mortal" (a universal negative proposition--E) must be false. If A is true, E is false. Another example: If "No men are dogs" (a universal negative proposition--E), then "Some men are dogs" (a particular affirmative proposition--I) must be false. The universal affirmative proposition--A) "All men are dogs," must also be false. The second rule states: "Contraries may be false together." If one contrary is false, the other contrary may also be false, although it need not be false, and may be true. Example: Consider the proposition "All men are content." This is a universal affirmative proposition (A) and let's consider this false. Since this statement is false, its contradictory, a particular negative proposition (O) "Some men are not content," must be true. But the Law of Subalternation states that the truth of the particular proposition does not involve the truth of the universal. Therefore, although it is true that "Some men are not content," we cannot validly conclude from this that its universal ("No men are content") is also true. "No men are content" (E) may be true or false. Therefore, both contraries may be false. The second rule is established. From the truth of one contrary we can conclude to the falsity of the other; but from the falsity of one contrary we cannot conclude to the truth of the other. The Law of Subcontrariety The Law of Subcontrariety: I--O. There are two rules to this law. The first rule states: "Both subcontraries cannot be false together." The first rule says:
Example: Consider the statement "Some men are dogs." Let's say that this particular affirmative (I) proposition is false. Since this is false, its contradictory (a universal negative--E) must be true, that is, "No men are dogs." If a universal proposition is true, its particular proposition is also true (the Law of Subalternation). So, since E is true, O must also be true and must state that "Some men are not dogs." If I is false, O is true. Another example: Let's say that O is false, that is, "Some men are not mortal." Its contradictory A, that is, "All men are mortal," must be true (the Law of Contradiction). But if A ("All men are mortal") is true, then I ("Some men are mortal"), must also be true (the Law of Subalternation). If O is false, I must be true. We can see now the truth of the first rule regarding subcontrary propositions (I and O). Subcontraries cannot be false together, at least one of the two must be true. The second rule of subcontraries (I and O) states: "Both subcontraries may be true together."
Example: Let's suppose it's true that "Some men are content," (a particular affirmative proposition--I). The contradictory of this proposition, "No men are content" (a universal negative proposition--E), must be false. We know, however, that the falsity of the universal does not involve the falsity of the particular (the Law of Subalternation). Therefore, even though E ("No men are content") is false, we cannot conclude that O ("Some men are not content") is false. This proposition may be true. Another example: Let's suppose that "Some men are not content" (a particular negative--O) is true. It's contradictory, "All men are content" (a universal affirmative--A), is false. We cannot conclude, however, from the falsity of the universal to the falsity of its particular (the Law of Subalternation), so it does not follow that I ("Some men are content") is also false. The statement "Some men are content" may be true. The two rules regarding subcontrary propositions (I and O) have now been established. Both subcontraries cannot be false together, but both subcontraries may be true together. Summary of the Laws We can now state the following conclusions:
The summary diagrammed:
To Page 2 of Opposition of Propositions
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